1. Field of the Invention
The present invention relates to a function generator which is supplied with an angular signal and produces function sin .phi. and cos .phi. or a function closely approximating a circular function. (In the case that a pair of functions X = f.sub.1 (.phi.) and Y = f.sub.2 (.phi.) are plotted along the X and Y axes of a rectangular coordinate system, if the locus of any pair of coordinates (X, Y) falls on a circle when the angle .phi. is linearly varied, the functions X = f.sub.1 (.phi.) and Y = f.sub.2 (.phi.) are referred to as circular functions in the present application).
2. Description of the Prior Art
In the hitherto known digital resolver converter or the like, a circuit which is supplied with an angular signal in a digital form and produces as an output a resolver signal in an analog form has been employed. For better understanding of the invention, a brief description will first be made of general arrangements and features of the heretofore known circuits employed for the digital resolver converter by referring to FIGS. 1 to 4.
FIG. 1 shows schematically a typical function generator employed for the digital resolver converter or the like. It is assumed that the function generator l shown in FIG. 1 is supplied with an angular signal .phi. of a digital quantity, whereby there appears at an output terminal 3 a function signal expressed by EQU X= f.sub.1 (.phi.).Es (1)
and at the other output the function is given by the following expression: EQU Y= f.sub.2 (.phi.).Es (2)
The angular signal .phi. applied to the input terminal 2 in a digital form comprising bits a.sub.0, a.sub.1, a.sub.2 , . . . , a.sub.n. The range in which the angular quantity applied to the input is varied is assumed to be 0.degree. to 90.degree., that is, within the first quadrant for the convenience' sake of the explanation. Further, it is also assumed that the weighting of the individual bits is made in accordance with the binary notation in a manner defined by the following expression: EQU .phi. = (a.sub.0 .times. 2.sup.0 + a.sub.1 .times. 2.sup.-.sup.1 + a.sub.2 .times. 2.sup.-.sup.2, . . ., a.sub.n .times. 2.sup.-.sup.n) .times. 45.degree. (3)
The range of .phi. given by the above expression (3) is of 0.degree. to 90.degree. .times. (1 - 2.sup.116 n.sup.-1).
By selecting n to be a sufficiently great number, .phi. can cover substantially the whole range from 0.degree. to 90.degree..
Reference numeral 5 indicates an input terminal for a reference signal Es required to generate the analog function. Application of the reference signal Es at the input terminal results in the generation of products of desired functions and Es at the outputs. In the case of the digital resolver converter or the like, an alternating current signal required for the excitation of the resolver is used as the reference signal Es. In other applications, a direct current signal may be used for the reference signal.
If the functions f.sub.1 (.phi.) and f.sub.2 (.phi.) having the following relation: ##EQU1## can be produced by the function generator 1, with a practically tolerable accuracy from the engineering viewpoint, the output signals X and Y may be satisfactorily utilized as the resolver signals. In this connection, it is noted that, even if the conditions that f.sub.1 (.phi.) .varies. cos .phi. and f.sub.2 (.phi.) .varies. sin .phi. are not always satisfied, the receiver servo system supplied with the above resolver signals X and Y can be operated normally, so far as the relation (4) is met, and the angular signal .phi. can be accepted with a practically tolerable accuracy.
As an example of such a function, the following functions ##EQU2## and ##EQU3## have been already reported in the periodical "ELECTRONIC DESIGN", Vol. 18, No. 7, Apr. 1, 1970, p. 56.
FIG. 2 shows a vector defined by the functions produced by the function generator shown in FIG. 1. The function f.sub.1 (.phi.) defined by the expression (5) is taken along the abscissa, while f.sub.2 (.phi.) satisfying the expression (6 ) is taken along the ordinate. In this case, the length of vector R and the angle .theta. can be, respectively, given by the following formulas: EQU R = .sqroot. F.sub.1.sup.2 (.phi.) + f.sub.2.sup.2 (.phi.) (7) ##EQU4##
If the constant K appearing in the formulas (5) and (6) is selected so that EQU K = 0.00617 (9)
and K' is so selected that R becomes equal to 1 when .phi. = 0, the vector length R will be varied in a manner shown in FIG. 3 as the signal .phi. varies from 0.degree. to 90.degree..
Further, if the term ##EQU5## of the expression (4) becomes ideally equal to tan .phi., the term .phi. of the expression (8) becomes equal to .phi.. However, in reality, the above condition can not be realized, and there arises an inevitable error between .theta. and .phi., which error will be varied in dependence on the values of the angle .phi. as is illustrated in FIG. 4.
When the functions given by the expressions (5) and (6) are utilized as X-axis signal and Y-axis signals, the angular error of the vector of these function signals is in the order of 0.032.degree. at maximum as can be seen from FIG. 4. Such errors lie in a tolerable range from the engineering viewpoint and hence the above functions can be satisfactorily utilized for the digital resolver converter and the digital synchro converter in practice.
The length of vector should ideally be constant independently from the angular signal .phi.. However, the vector length is decreased about 14 % at maximum at 45.degree. as is illustrated in FIG. 3. Although such variation may be tolerated in the case of the digital resolver converter or the like under certain circumstances, it can be allowed in the other applications such as display devices or the like.
As will be appreciated from the foregoing discussion, the function generator which can produce the functions defined by the expressions (5) and (6) has a drawback that the vector length of the functions is subjected to variations in dependence upon the angular values and therefore can not be called an ideal circular function generator. In other words, the functions expressed by the formulas (5) and (6) will certainly satisfy the condition given by the expression (4) in respect of the angular value with a high accuracy. These functions, however, are not proportional to cos .phi. and sin .phi. and for this reason incurs a result that the vector length will not remain constant.